Mexico's former Arsenal striker Carlos Vela cancelled out Bradley's sixth-minute chip to give the hosts a point.
Mexico's former Arsenal striker Carlos Vela cancelled out Bradley's sixth-minute chip to give the hosts a point which saw them extend their lead at the top of the CONCACAF standings to six points.
The USA, who made seven changes from the team which beat Trinidad & Tobago in Colorado last Thursday, are level with second-place Costa Rica on eight points but trail on goal difference.
The top three teams in the six-team round robin competition qualify automatically for next year's finals in Russia.
US skipper Bradley later praised the American resilience as they dug deep for a vital point.
"Obviously, they had a lot of the ball but tactically the game went the way we expected and the way we wanted," Bradley said.
"Anytime you can get a point here, it's great. The commitment and the mentality from every guy was really, really good and hopefully we can keep moving this team forward."
Bradley also had words of praise for US coach Bruce Arena, who has helped steady the US qualifying campaign since returning to the helm after the sacking of predecessor Jurgen Klinsmann last year.
"Look, we were pretty honest with ourselves and understood that we had let a lot of little things drop," Bradley said.
"Bruce came in and has done a really good job of little by little making sure we get back to who we are. Now we just have to keep going."
The tone for a frenetic first half at the Estadio Azteca was set in the opening minutes, when Mexico defender Carlos Salcedo was lucky not to see a red flag after thrusting an elbow into Bobby Wood's face.
But soon Mexico were rocked by a superb strike from Bradley which stunned the Azteca arena into silence.
With the US pressing high up the field, Bradley pounced on a loose pass near halfway. The US captain glanced up to spot Guillermo Ochoa off his line and deftly chipped from 40 yards out for a wonderful goal.
Mexico responded swiftly, with Jonathan Dos Santos firing a low shot past Brad Guzan's left-hand post.
Javier Hernandez then triggered alarm by springing the US offside trap to go through on goal after 20 minutes.
Guzan was alert to the danger, however, and was swiftly off his line to off-balance Hernandez, whose looping effort was cleared to safety by Omar Gonzalez.
Three minutes later Mexico got their equaliser following a lightning counter-attack after Wood had miscued a close-range effort. Ochoa's quick throw found Hernandez, who danced clear of his marker before feeding Vela.
The striker cut in from the right flank, surged past DaMarcus Beasley and rifled a shot into the bottom corner of Guzan's goal.
Despite the Mexican pressure, the US continued to create chances, the best of which fell to Gonzalez who should have done better with a 39th-minute header that went wide with the goal gaping.
Both sides maintained the furious pace in the second half, with Mexico pressing relentlessly in search of a goal.
The US defence, with Stoke City centre-half Geoff Cameron outstanding, was able to withstand the Mexican pressure to the frustration of the home crowd.
The best Mexico chance came in the 71st minute, when Hector Herrera smashed a venomous free-kick onto the bar.
The US continued to look lively on the counter-attack, and Bradley almost added another wonder-goal to his account on 74 minutes with a dipping volley from 30 yards that hit the post.